In order to simplify the compilation process, many compilers for higher-order languages use the continuation-

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Continuationsemantics is the traditional mathematical formalism for specifying the semantics of non-local control operations.Modern Lisp-style languages, however, contain advanced control structures like full functional jumps and control delimiters for which continuation semantics is insufficient. We s,olve this problem by introducing an abstract domain of rests of computations with appropriate operations. Beyond being useful for the problem at hand, these abstract continuations turn out to have applications in a much broader context, e.g., the explication of parallelism, the modeling of control facilities in parallel languages, and the design of new control structures.

In order to simplify the compilation process, many compilers for higher-order languages use the continuation-passing style (CPS) transformation in a first phase to generate an intermediate representation of the source program. The salient aspect of this intermediate form is that all procedures take an argument that represents the rest of the computation (the "continuation"). Since the naïve CPS transformation considerably increases the size of programs, CPS compilers perform reductions to produce a more compact intermediate representation. Although often implemented as a part of the CPS transformation, this step is conceptually a second phase. Finally, code generators for typical CPS compilers treat continuations specially in order to optimize the interpretation of continuation parameters.A thorough analysis of the abstract machine for CPS terms shows that the actions of the code generator invert the naïve CPS translation step. Put differently, the combined effect of the three phases is equivalent to a source-to-source transformation that simulates the compaction phase. Thus, fully developed CPS compilers do not need to employ the CPS transformation but can achieve the same results with a simple source-level transformation.

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